A general view shows a camp where detained ISIS families are held in Ain Issa in the Syrian northern Kurdish region on February 15, 2018. Delil Souleiman / AFP

Fighters from ISIS began leaving areas near Damascus on Sunday under an evacuation deal as Kurdish-led forces launched an attack on the extremist group in eastern Syria.

The departure of the militants and their families from the Yarmouk Palestinian refugee camp and adjacent Al Tadamun neighbourhood south of Damascus brings the Syrian government closer to full control of the capital region for the first time since the civil war broke out in 2011.

The evacuation was reported by the Syrian Observatory for Human Rights, a Britain-based war monitor, which said the deal was negotiated by Syrian regime ally Russia and Palestinian factions that have been fighting the militants alongside government forces.

The Observatory said buses had arrived after midnight to take out the ISIS fighters and their families. Observatory head Rami Abdel Rahman said the militants burned their headquarters in Yarmouk before boarding the buses with their relatives.

"The six buses left at dawn, heading east for the Syrian desert," he told Agence France-Presse.

However, a military source quoted by Syria's state-run Sana news agency denied there had been any agreement, and the the state-run Al Ikhbariya TV reported that government forces were storming ISIS positions in Hajar Al Aswad, also near Yarmouk.

President Bashar Al Assad's forces launched an offensive against ISIS in Yarmouk and Hajar Al Aswad last month after capturing the Eastern Ghouta rebel stronghold near the capital. Both sides have suffered heavy casualties, with 484 ISIS militants, more than 250 pro-regime fighters and 56 civilians killed so far, according to Observatory.

The fighting subsided on Saturday amid reports that an evacuation deal was being negotiated.

Abdullah Al Khateeb, a human-rights activist in the Yalda neighbourhood less than two kilometres from Yarmouk, confirmed that talks were held on an evacuation deal. However, "the issue of where the extremists will be taken remains disputed", he told The National.

Government forces gained control of more than 70 per cent of Yarmouk in the past month, but many civilians remain trapped inside, Mr Al Khateeb said.

"There are around 200 civilians are trapped inside the camp, they cannot leave. The media has not been able to cover reports that hundreds of people including, women and children have been killed inside the camp. We are unable get the dead bodies out," he said.

"There are no doctors, nurses, ambulances inside of the camp. There are no water supplies. Government troops have completely besieged the area."

The United Nations has repeatedly warned of the "catastrophic consequences" of the fighting on civilians trapped in Yarmouk, which began as a refugee camp for Palestinians who fled or were expelled from their homes in the 1948 war leading to the creation of Israel. On the eve of Syria's civil war it was a built-up residential area home to about 160,000 Palestinians and Syrians.

In eastern Syria, an alliance of Kurdish and Arab fighters advanced against ISIS near the border with Iraq, the Observatory said.

The Syrian Democratic Forces (SDF), backed by US and French forces, captured a strategic hilltop in Hajjin village, one of the three villages ISIS controls in Deir Ezzor province bordering Iraq.

"There are intense clashes around Hajjin, and the SDF is advancing thanks to American and French artillery fire," Mr Abdel Rahman said. The international anti-ISIS coalition forces had been backing the SDF's operation with air strikes but switched to artillery as they drew closer to the insurgents, he said.

The coalition of Kurdish and Arab militias had paused their fight against ISIS after Turkey launched an assault in January against their northern region of Afrin.

The terror group has been driven from virtually all the territory it once controlled in Syria and Iraq, but is still present in remote areas along the border.

Syria's war has left more than 350,000 people dead since it started in 2011 with the brutal repression of anti-government protests and spiralled into a complex conflict involving world powers and an array of rebel and extremist groups.